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Theorem csbriotag 6317
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
csbriotag  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph ) )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)    V( x, y)

Proof of Theorem csbriotag
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3084 . . 3  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota_ y  e.  B ph )  =  [_ A  /  x ]_ ( iota_ y  e.  B ph ) )
2 dfsbcq2 2994 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32riotabidv 6306 . . 3  |-  ( z  =  A  ->  ( iota_ y  e.  B [
z  /  x ] ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph )
)
41, 3eqeq12d 2297 . 2  |-  ( z  =  A  ->  ( [_ z  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [
z  /  x ] ph )  <->  [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph )
) )
5 vex 2791 . . 3  |-  z  e. 
_V
6 nfs1v 2045 . . . 4  |-  F/ x [ z  /  x ] ph
7 nfcv 2419 . . . 4  |-  F/_ x B
86, 7nfriota 6314 . . 3  |-  F/_ x
( iota_ y  e.  B [ z  /  x ] ph )
9 sbequ12 1860 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
109riotabidv 6306 . . 3  |-  ( x  =  z  ->  ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [ z  /  x ] ph ) )
115, 8, 10csbief 3122 . 2  |-  [_ z  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [ z  /  x ] ph )
124, 11vtoclg 2843 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   [wsb 1629    e. wcel 1684   [.wsbc 2991   [_csb 3081   iota_crio 6297
This theorem is referenced by:  cdlemkid3N  31122  cdlemkid4  31123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-riota 6304
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